In which circumstance will there be a substantial difference between the variable's mean and median values?

Question

Options

• AWhen the variable is of the categorical type
• BWhen the variable is of the boolean type
• CWhen the variable contains no outliers
• DWhen the variable contains a lot of extreme outliers

• A
  14% (4)
• B
  4% (1)
• C
  4% (1)
• D
  79% (22)

Explanation

D is correct because extreme outliers pull the mean toward them (since mean uses every value in its calculation), while the median - being the middle value - remains resistant to outliers. A heavily skewed distribution, like household incomes where a few billionaires exist, will have a mean far higher than the median.

Why the distractors are wrong:

• A (categorical): Mean and median aren't typically meaningful for categorical variables at all - the comparison is irrelevant.
• B (boolean): A boolean (0/1) variable can have a mean (proportion of 1s) and a median, but these won't substantially differ in unusual ways beyond what any skewed distribution causes.
• C (no outliers): Without outliers, the distribution tends to be more symmetric, so mean and median stay close to each other - the opposite of what the question asks.

Memory tip: Think of the mean as a "sensitive" measure (easily disturbed by extremes) and the median as a "stubborn" one (ignores extremes). A big gap between them is a red flag that outliers are dragging the mean away - statisticians call this skewness.

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